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( 3 Hours)
[ Total
La) Design the finite state mpchine that compares two binarr numbers to
determine whetherthey are equal and which of the two I~ Ic1..Yj fJ-"",
b)(i)State.and prove the pumping lemma for context nee languages.
(ii)Convertthe following regular expressionto NFA with e transitiQns.
R= (1(00)*1+01.*0)*
7.a) Prove the following:
-
(08)
(08)
(04)
(10)
(i) L = {aP I p is prime}is notcontextnee.
(ii) L = { (ab) n ak : n>k, k~O} is not regular.
b) Consider the following grammar:
S~ASB
-
(10)
.
e
I
A -taAS a
I
B-tSb& A bb
Put the above grammar in Chomskynormal form.
I
..,
} tJ (; .
TV-8307
Marks:
(1) '()Jest ion No.1 is coIIpllsory.
(2) A~tempt any four questions
out of remaining six,questions.
(3) Assumptions made should be clearly
stated.
(4) ,Ass':1IIle suitable
data wherever required
but justify
the same.
(5) Fi<J1:1res to the right indicate
full marks.
,
r{'
[ REVISED COORSE ]
,A .
.
v/:tJ
I
J. a) Give the DFA accepting the following language over alphabet {O,I }
(08)
L = 'Set of all strings beginning with I that, when interpreted as a binary
integer, is a multiple of5: for example, strings 101,1010, and 1111 in the
language; 0, 100, and t II not.
b) (i) Design turing machille that can accept the set of all even palindromes (08)
over alphabet {0,1}
(ii) Show that that
(04)
(1+00*1)+ (1+00*1)(0+10*1) * (OHO*l) = 0*1(0+ 10*1) *
4. a) Construct the PDA equivalent to the following context free grammar.
(10)
S-+ OaB
B-t OS lIS I 0
Test whether 0104 is in language.
b)Provethat it is undecidablewhether a context free grarnmaris ambiguous.(IO)
5. a) Convertthe following gramrnarinto Grebaicnormal form.
. S-t XY1 I 0
X~ OOX Y
Y--IXI
(10)
I
I
b) Design turing machine to recognize the language L = {I n2n3n n2:I} (10)
6. a) Designthe PDA that will recognizethe language L = WWR: W is in (10)
{a,b}* i,e even length palindrome over I = {a,b}.
b) Give the moore and mealy machinefor the following
(10)
For the input nom l:*, where L = {0,1,2},printthe residu modulo 5
of the input treated as ternary..
100
(06)
7.a) Write regular expressions for the following languages
i)'L = {O,l}containing all possible combinations of D's and l's but
not having two consecutive O's.
ii) The set of all strings ofO's and l's such that every pair of adjacent O's
appears before any pair of adjacent 1'so
iii)The set of all strings over L = {O,l}without length two.
. (06)
b) i) Verify the following identities involving regular expressions.
.
(R+S)+T = R+(S+T)
(RS)T = R(ST)
ii) Write short notes on any two
(i)
Grebaic Theorm.
(ii)
P<?stcorrespondence problem.
(iii)
Myhill-Nerode's Theorm.
(08)